On exponentiation of \(G\)-sets
The author presents two theorems about exponentiation of (actual or virtual) \(G\)-sets, where \(G\) is a finite group. The first theorem uses a special case of Dress's concept of exponentiation for virtual \(G\)-sets (cf. \textit{A. Dress} [Proc. Symp. Pure Math. 21, 39-45 (1971; Zbl 0248.20007)]) to clarify the rule of signs introduced by \textit{A. Joyal} [C. R. Math. Acad. Sci., Soc. R. Can. 7, 285-290 (1985; Zbl 0605.20009)] in his theory of virtual sets. The second theorem relates the power sets of (actual) \(G\)-sets to the corresponding linear representations of \(G\): For any two \(G\)-sets \(A\) and \(B\), the following are equivalent. (a) \({\mathbf C} A\) and \({\mathbf C} B\) are equivalent representations of \(G\). (b) \(2^{A}\) and \(2^{B}\) are isomorphic \(G\)-sets. (Here 2 is a two- element set with trivial \(G\)-action, so \(2^ A\) is the power set of \(A\).) (c) \({\mathbf C}2^ A\) and \({\mathbf C} 2^ B\) are equivalent representations of \(G\). The paper contains the proofs of the theorems and some additional remarks and examples.
scientific article; zbMATH DE number 716447